I feel like there's a lot of ambiguity here, but I guess that's sometimes the nature of Japanese, and immediate understanding of what part of the relative clause the noun refers to would come with fluency.

Anyway, is this right? What general rules are there for nouns referring to different parts of the relative clause?

下を走った橋 should be understandable if there's enough context, but if you worry about ambiguity, it may be better to explicitly add the subject of 走る and say 私が下を走った橋. Although we don't usually say 橋が走る, we do commonly say 道路/川が走る "the street/river runs (e.g., across the town)", and thus 下を走った橋 without any other modifier might also be taken as "a bridge that crosses some low place" by some people.